A natural resource such as oil or gas residing in a subterranean formation can be recovered by drilling a well into the formation. The subterranean formation is usually isolated from other formations using a technique known as cementing. In particular, a well bore is typically drilled down to the subterranean formation while circulating a drilling fluid through the well bore. After the drilling is terminated, a string of pipe (e.g. casing string) is run in the well bore. Primary cementing is then usually performed whereby a cement slurry is pumped down through the casing string and into the annulus between the casing string and the wall of the well bore or another casing string to allow the cement slurry to set into an impermeable cement column and thereby fill a portion of the annulus. Sealing the annulus typically occurs near the end of cementing operations after well completion fluids, such as spacer fluids and cements, are trapped in place to isolate these fluids within the annulus from areas outside the annulus. The annulus is conventionally sealed by closing a valve, energizing a seal, and the like.
After completion of the cementing operations, production of the oil or gas may commence. The oil and gas are produced at the surface after flowing through the casing string. As the oil and gas pass through the casing string, heat may be passed from such fluids through the casing string into the annulus. As a result, thermal expansion of the fluids in the annulus above the cement column causes an increase in pressure within the annulus also known as annulus pressure buildup. Annulus pressure buildup typically occurs because the annulus is sealed and its volume is fixed. Annulus pressure buildup may cause damage to the well bore such as damage to the cement sheath, the casing, tubulars, and other equipment. In addition, annulus pressure buildup makes proper casing design difficult if not impossible. Because the fluid pressures may be different in the annulus for each well bore, use of a standard casing design may not be practical. In order to control annulus pressure buildup, conventional methods circulate gas into place during cementing operations. Because the gas is mobile, it is difficult to place the gas in the proper location and, at the same time, control the fluid pressure in the annulus. If, for example, the gas is placed too far below the top of the annulus, the rising gas will increase the pressure in the annulus.
In order to maintain a safe and acceptable pressure within the sealed annulus, the pressure within the sealed annulus must be calculated within some level of certainty. Some methods have been proposed for determining annulus pressure buildup, which are based on volume changes. One method, for example, calculates the fluid volume change ΔVf using the fluid bulk modulus K and the volume coefficient of thermal expansion β according to equation (1):
                              Δ          ⁢                                          ⁢                      V            f                          =                              V            f                    ⁡                      (                                          βΔ                ⁢                                                                  ⁢                T                            -                                                Δ                  ⁢                                                                          ⁢                  P                                K                                      )                                              (        1        )            where Vf is the fluid volume, ΔT is the temperature change, and ΔP is the pressure change. Equation (1) uses parameters that are derived from the PVT (pressure-volume-temperature) behavior of the fluid in the sealed annulus, not properties that are directly measured. Because equation (1) applies to the entire annulus, the values of K and β represent some type of average value that must be properly chosen to get the correct answer. And, equation (1) may be used to calculate the pressure within the sealed annulus within some level of certainty only for sufficiently small values of ΔT and ΔP. Another method directly uses the PVT behavior of the fluid and integrates the volume change over the length of the annulus to calculate the fluid volume change ΔVf according to equation (2):
                              Δ          ⁢                                          ⁢                      V            f                          =                  -                                    ∫                              s                0                                            s                1                                      ⁢                                          Δρ                                  ρ                  +                                      Δ                    ⁢                                                                                  ⁢                    ρ                                                              ⁢              A              ⁢                                                          ⁢              ds                                                          (        2        )            where ρ is the initial density, βρ is the change in density, A is the annulus cross-sectional area and s is the axial coordinate (measured depth) of the annulus. The assumption in equation (2) is that the temperature and pressure are functions of s, and that density is calculated as a function of temperature and pressure, so that the integrand of equation (2) also varies with s. In both methods represented by equations (1) and (2), the casing volume change ΔVc is calculated as a function of pressure and temperature using Lame's equation from conventional elasticity theory. The annulus pressure buildup ΔPb for either the method represented by equation (1) or the method represented by equation (2) is thus, determined when:ΔVf=ΔVc  (3)
The disadvantage of both methods is apparent when multiple fluids are present in the sealed annulus. A gas, for example, may be introduced at the base of the annulus, which later migrates to the top of the annulus. This gas cannot be characterized by its volume according to the foregoing methods because its volume varies with the pressure and temperature. In other words, gas at the base of the annulus will usually have a higher temperature and pressure than gas at the top of the annulus.